Properties

Label 3.33.b.a
Level $3$
Weight $33$
Character orbit 3.b
Analytic conductor $19.460$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,33,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 33, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 33);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.4599965427\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 954745942 x^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{61}\cdot 5^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 35 \beta_1 - 2138715) q^{3} + (\beta_{3} + 18 \beta_{2} + 5 \beta_1 - 2579203486) q^{4} + ( - \beta_{4} + \beta_{3} - 1092 \beta_{2} + 101591 \beta_1) q^{5} + (\beta_{6} - 4 \beta_{4} + 141 \beta_{3} + 619 \beta_{2} + \cdots - 242453078830) q^{6}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + 10 \beta_{7} + \cdots - 79012360393819) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 35 \beta_1 - 2138715) q^{3} + (\beta_{3} + 18 \beta_{2} + 5 \beta_1 - 2579203486) q^{4} + ( - \beta_{4} + \beta_{3} - 1092 \beta_{2} + 101591 \beta_1) q^{5} + (\beta_{6} - 4 \beta_{4} + 141 \beta_{3} + 619 \beta_{2} + \cdots - 242453078830) q^{6}+ \cdots + ( - 16\!\cdots\!04 \beta_{9} + \cdots - 33\!\cdots\!52) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 21387150 q^{3} - 25792034864 q^{4} - 2424530788848 q^{6} - 5568062418940 q^{7} - 790123604155542 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 21387150 q^{3} - 25792034864 q^{4} - 2424530788848 q^{6} - 5568062418940 q^{7} - 790123604155542 q^{9} - 70\!\cdots\!40 q^{10}+ \cdots - 33\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 954745942 x^{8} + \cdots + 31\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 59\!\cdots\!15 \nu^{9} + \cdots + 39\!\cdots\!20 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!15 \nu^{9} + \cdots - 16\!\cdots\!40 ) / 55\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 67\!\cdots\!31 \nu^{9} + \cdots - 18\!\cdots\!40 ) / 41\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 72\!\cdots\!03 \nu^{9} + \cdots - 39\!\cdots\!20 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 73\!\cdots\!33 \nu^{9} + \cdots - 12\!\cdots\!00 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38\!\cdots\!55 \nu^{9} + \cdots + 20\!\cdots\!20 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 99\!\cdots\!25 \nu^{9} + \cdots - 88\!\cdots\!80 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!21 \nu^{9} + \cdots + 66\!\cdots\!00 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 18\beta_{2} + 5\beta _1 - 6874170782 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 4 \beta_{7} - 40 \beta_{6} - \beta_{5} + 172 \beta_{4} - 1896 \beta_{3} + 1835342 \beta_{2} - 11560394524 \beta _1 - 688 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1827 \beta_{9} - 1827 \beta_{8} - 42637 \beta_{7} - 535448 \beta_{6} - 15095 \beta_{5} + 1492743 \beta_{4} - 2123819903 \beta_{3} - 57318530480 \beta_{2} + \cdots + 99\!\cdots\!16 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 101370960 \beta_{9} - 2295545507 \beta_{8} + 20199018044 \beta_{7} + 119767606616 \beta_{6} + 2701029347 \beta_{5} - 1029997038516 \beta_{4} + \cdots + 2549541198128 ) / 972 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2013405595299 \beta_{9} + 2013405595299 \beta_{8} + 54481961761549 \beta_{7} + 612483446138776 \beta_{6} + 1725271115895 \beta_{5} + \cdots - 57\!\cdots\!36 ) / 243 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 40\!\cdots\!60 \beta_{9} + \cdots - 71\!\cdots\!64 ) / 486 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11\!\cdots\!54 \beta_{9} + \cdots + 23\!\cdots\!04 ) / 243 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11\!\cdots\!00 \beta_{9} + \cdots + 17\!\cdots\!84 ) / 243 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
21147.4i
17303.1i
9879.94i
9002.40i
5429.47i
5429.47i
9002.40i
9879.94i
17303.1i
21147.4i
126884.i 2.43525e7 3.54961e7i −1.18046e10 1.55775e11i −4.50390e12 3.08995e12i 3.32502e12 9.52859e14i −6.66929e14 1.72884e15i −1.97653e16
2.2 103819.i −2.32547e7 + 3.62248e7i −6.48340e9 2.49832e11i 3.76082e12 + 2.41428e12i −3.04380e13 2.27200e14i −7.71458e14 1.68480e15i 2.59372e16
2.3 59279.7i −4.29008e7 3.54120e6i 7.80890e8 2.23415e11i −2.09921e11 + 2.54315e12i 4.13333e13 3.00895e14i 1.82794e15 + 3.03841e14i −1.32440e16
2.4 54014.4i 3.76547e7 + 2.08602e7i 1.37741e9 5.87306e9i 1.12675e12 2.03389e12i 1.74600e13 3.06390e14i 9.82725e14 + 1.57097e15i −3.17230e14
2.5 32576.8i −6.54523e6 4.25462e7i 3.23372e9 1.19330e11i −1.38602e12 + 2.13223e11i −3.44644e13 2.45260e14i −1.76734e15 + 5.56949e14i 3.88740e15
2.6 32576.8i −6.54523e6 + 4.25462e7i 3.23372e9 1.19330e11i −1.38602e12 2.13223e11i −3.44644e13 2.45260e14i −1.76734e15 5.56949e14i 3.88740e15
2.7 54014.4i 3.76547e7 2.08602e7i 1.37741e9 5.87306e9i 1.12675e12 + 2.03389e12i 1.74600e13 3.06390e14i 9.82725e14 1.57097e15i −3.17230e14
2.8 59279.7i −4.29008e7 + 3.54120e6i 7.80890e8 2.23415e11i −2.09921e11 2.54315e12i 4.13333e13 3.00895e14i 1.82794e15 3.03841e14i −1.32440e16
2.9 103819.i −2.32547e7 3.62248e7i −6.48340e9 2.49832e11i 3.76082e12 2.41428e12i −3.04380e13 2.27200e14i −7.71458e14 + 1.68480e15i 2.59372e16
2.10 126884.i 2.43525e7 + 3.54961e7i −1.18046e10 1.55775e11i −4.50390e12 + 3.08995e12i 3.32502e12 9.52859e14i −6.66929e14 + 1.72884e15i −1.97653e16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.33.b.a 10
3.b odd 2 1 inner 3.33.b.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.33.b.a 10 1.a even 1 1 trivial
3.33.b.a 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{33}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 34370853912 T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{10} + 21387150 T^{9} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + 2784031209470 T^{4} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 67\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 94\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
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